Illiquidity and Derivative Valuation

Ulrich Horst, Felix Naujokat

Institut für Mathematik

Humboldt-Universität Berlin

horst@mathematik.hu-berlin.de

naujokat@mathematik.hu-berlin.de

November 15, 2021

Abstract

In illiquid markets, option traders may have an incentive to increase their portfolio value

by using their impact on the dynamics of the underlying. We provide a mathematical frame-

work within which to value derivatives under market impact in a multi-player framework by

introducing strategic interactions into the Almgren & Chriss (2001) model. Speci�cally, we

consider a �nancial market model with several strategically interacting players that hold Eu-

ropean contingent claims and whose trading decisions have an impact on the price evolution of

the underlying. We establish existence and uniqueness of equilibrium results and show that the

equilibrium dynamics can be characterized in terms of a coupled system of possibly non-linear

PDEs. For the linear cost function used in Almgren & Chriss (2001), we obtain (semi) closed

form solutions for risk neutral or CARA investors. Finally, we indicate how spread crossing

costs discourage market manipulation.

Preliminary Version - Comments Welcome

AMS classi�cation: 91B28, 91B70, 60K10

JEL classi�cation: C73, G12, G13

Keywords: Stochastic di�erential games, illiquidity, market impact, derivative valuation.

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1 Introduction

Standard �nancial market models assume that asset prices follow an exogenous stochastic process

and that all transactions can be settled at the prevailing price without any impact on market

dynamics. The assumption that all trades can be carried out at exogenously given prices is

appropriate for small investors that trade only a negligible proportion of the overall daily trading

volume; it is not appropriate for institutional investors trading large blocks of shares over a short

time span. The trading decisions of institutional investors are likely to move stock prices in an

unfavorable direction and often induce signi�cant trading costs.

It is now widely recognized that (the lack of) liquidity is a major source of �nancial risk

and there has been an increasing interest in mathematical models of illiquid �nancial markets.

Much of the literature on illiquidity focusses on either optimal hedging and portfolio liquidation

strategies for a single large investor under market impact (Cetin, Jarrow & Protter (2004), Alfonsi,

Fruth & Schied (2007), Rogers & Singh (2007)), predatory trading (Carlin, Lobo & Viswanathan

(2007), Schoeneborn & Schied (2008)) and the role of derivative securities including the problem

of market manipulation using options (Jarrow (1992), Kumar & Seppi (1992)). In an illiquid

market, derivative traders have an incentive to utilize their impact on the price dynamics of the

underlying in order move the option value in a favorable direction1. It has been shown by Jarrow

(1994), for instance, that by introducing derivatives into an otherwise complete and arbitrage-free

market, manipulation strategies with a risk free gain may appear, such as market corners and front

runs. Schoenbucher & Wilmott (2000) discuss an illiquid market model where a large trader can

in�uence the stock price with vanishing costs and risk. They argue that the risk of manipulation

on the part of the large trader makes the small traders unwilling to trade derivatives any more. In

particular, they predict that the option market breaks down. Our analysis indicates that markets

do not necessarily break down when stock price manipulation is costly as it is in our model.

While the aforementioned papers di�er signi�cantly in their degree of complexity, they all focus

on a single player framework. When multiple players are considered the analysis is typically con-

�ned to some form of �stealth trading� as in Carlin, Lobo & Viswanathan (2007) and Schoeneborn

& Schied (2008), where liquidity providers try to bene�t from the liquidity demand that comes

from some �large� investor but no strategic interaction between liquidity suppliers and consumers

is considered. In fact, so far only little work has been devoted to models with strategically in-

teracting market participants. Vanden (2005) considers a pricing game in continuous time where

the option issuer controls the volatility of the underlying but does not incur liquidity or spread

crossing costs. He derives a Nash equilibrium in the two player, risk neutral case and shows that

�seemingly harmless derivatives, such as ordinary bull spreads, o�er incentives for manipulation

that are identical to those o�ered by digital options� (p. 1892, l. 36). Gallmeyer & Seppi (2000)

1Gallmeyer & Seppi (2000) provide some evidence that in illiquid markets option traders are in fact able to

increase a derivative's value by moving the price of the underlying.

2

consider a binomial model with three periods and �nitely many risk neutral agents holding call

options on an illiquid underlying. Assuming a linear permanent price impact and linear transac-

tion costs, and assuming that all agents are initially endowed with the same derivative they prove

the existence of Nash equilibrium trading strategy and indicate how market manipulation can be

reduced.

We provide a general mathematical framework within which to value derivative securities in

illiquid markets under strategic interactions thereby extending the work of Gallmeyer & Seppi

(2000) in several directions. Speci�cally, we consider a pricing game between a �nite number of

large investors (�players�) holding European claims written on an illiquid stock. Their goal is

to maximize expected utility at maturity from trading the stock where their portfolio value at

maturity depends on the trading strategies of all the other players, due to their impact on the

dynamics of the underlying. Following Almgren & Chriss (2001) we assume that the players have

a permanent impact on stock prices and that all trades are settled at the prevailing market price

plus a liquidity premium. The liquidity premium can be viewed as an instantaneous price impact

that a�ects transaction prices but not the value of the players' inventory. This form of market

impact modeling is analytically more tractable than that of Obizhaeva & Wang (2006) which

also allows for temporary price impacts and resilience e�ects. It has also been adopted by, e.g.,

Carlin, Lobo & Viswanathan (2007), Schoeneborn & Schied (2008) and many practitioners from

the �nancial industry.

Our framework is �exible enough to allow for rather general liquidity costs including the linear

cost function of Almgren & Chriss (2001) and some form of spread crossing costs. We show

that when the market participants are risk neutral or have CARA utility functions the pricing

game has a unique Nash equilibrium in the class of absolutely continuous trading strategies;

existence results for more general utility functions are given for the one player case. We solve

the problem of equilibrium pricing using techniques from the theory of stochastic optimal control

and stochastic di�erential games. We show that the family of the players' value functions can

be characterized as the solution to a coupled system of non-linear PDEs. Here we use a-priori

estimates for Nash equilibria; we prove that the system of PDEs has a unique classical solution

with bounded derivatives. It turns out that the equilibrium problem can be solved in closed form

for a speci�c market environment, namely the linear cost structure used in Almgren & Chriss

(2001) and risk neutral agents. We use this explicit solutions to state some conditions which make

manipulation unattractive or avoid it altogether. For instance, we show that when the agents

are risk neutral no market manipulation occurs in zero sum games, i.e., in a game between an

option writer and an option issuer. Furthermore, we �nd that the bid ask spread is important

determinant of market manipulation. It turns out that the higher the spread, the less bene�cial

market manipulation: high spread crossing costs make trading more costly and hence discourage

frequent re-balancing of portfolio positions.

3

This paper is organized as follows: We present the market model in section 2. In chapter 3,

we formulate the optimization problem, derive a priori estimates for Nash equilibria and prove the

existence of a solution for one player with general utility function. We solve the multi-player case

in section 4 for risk neutral and CARA agents. We use these solutions in chapter 5 to show how

market manipulation can be reduced. Section 6 concludes.

2 The Model

We adopt the market impact model of Schoeneborn & Schied (2007) with a �nite set J of agents,

or players, trading a single stock whose price process depends on the agent's trading strategies.

Following Almgren & Chriss (2001) we shall assume that the players have a permanent impact on

asset prices and that all trades are settled at prevailing market prices plus a liquidity premium

which depends on the change in the players' portfolios. In order to be able to capture changes

in portfolio positions in an analytically tractable way, we follow Almgren & Chriss (2001) and

Schoeneborn & Schied (2007) and restrict ourselves to absolutely continuous trading strategies.

Hence we consider only trading strategies from the class

X , {X : [0, T ] 7→ R|X absolutely continuous, adapted and X0 = 0}

where we denote by Xjt the number of stock shares held by player j ∈ J , {1, ..., N} at time

t ∈ [0, T ]. We write dXjt = Xj

t dt and call Xj the trading speed of the player j.

2.1 Price dynamics and the liquidity premium

Our focus is on valuation schemes for derivatives with short maturities under strategic market

interactions. For short trading periods it is appropriate to model the fundamental stock price, i.e.,

the value of the stock in the absence of any market impact, as a Brownian Motion with volatility

(σBt). Market impact is accounted for by assuming that the investors' accumulated stock holdings∑Ni=1X

i have a linear impact on the stock process (Pt) so that

Pt = P0 + σBt + λN∑i=1

Xit (2.1)

with a permanent impact parameter λ > 0. The linear permanent impact is consistent with the

work of Huberman & Stanzl (2004) who argued that linearity of the permanent price impact is

important to exclude quasi-arbitrage2.

2There is some empirical evidence that very large trades have a concave price impact but this observation needs

further validation.

4

A trade at time t ∈ [0, T ] is settled at a transaction price Pt that includes an additional

instantaneous price impact, or liquidity premium. Speci�cally,

Pt = Pt + g

(N∑i=1

Xit

)(2.2)

with a cost function g that depends on the instantaneous change∑N

i=1 Xi in the agents' position

in a possibly non-linear manner. The liquidity premium accounts for limited available liquidity,

transaction costs, fees, spread crossing costs, etc. Spread-crossing costs are of particular impor-

tance and have not been considered in the previous literature on market impact.

Remark 2.1. In our model the liquidity costs are the same for all traders and depend only on the

aggregate demand throughout the entire set of agents. This captures situations where the agents

trade through a market maker or clearing house that reduces the trading costs by collecting all orders

and matching incoming demand and supply prior to settling the outstanding balance∑N

i=1 Xit at

market prices. �

We assume with no loss of generality that g is normalized, i.e., g(0) = 0 and smooth. The

following additional mild assumptions on g will guarantee that the equilibrium pricing problem

has a solution for risk neutral and CARA investors.

Assumption 2.2. • The derivative g′ is bounded away from zero, that is g′ > ε > 0.

• The mapping z 7→ g(z) + zg′(z) is strictly increasing.

The �rst condition is natural for a cost function. Since the liquidity costs associated with a net

change in the overall position z is given by zg(z), the second assumption states that the agents

face increasing marginal costs of trading. Our assumptions on g are satis�ed for the following

important examples:

Example 2.3. Cost functions which satisfy Assumption 2.2 are the linear cost function g(z) = κz

with κ > 0, used in Almgren & Chriss (2001) and cost functions of the form

g(z) = κz + s2π

arctan(Cz) with s, C > 0.

The former is the cost function associated with a block-shaped limit order book. The latter can

be viewed as a smooth approximation of the map z 7→ κz + s · sign(z) which is the cost function

associated with a block-shaped limit order book and spread s > 0.

2.2 Preferences and endowments

Each agent is initially endowed with a contingent claim Hj = Hj(PT ), whose payo� depends on

the stock price PT at maturity. Although it is not always necessary we assume that the functions

Hj are smooth and bounded with bounded derivatives Hjp .

5

Remark 2.4. We only consider options with cash delivery. The assumption of cash delivery is

key. While cash settlement is susceptible to market manipulation, we show in Section 5 below

that when deals are settled physically, i.e., when the option issuer delivers the underlying, market

manipulation is not bene�cial: the cost of acquiring at an increased price outweighs the bene�ts

from a possible higher option payo�, due to an increase in the underlying. �

We model the risk preference of the agent j ∈ J with a von Neumann - Morgenstern utility

function uj and assume that her aim is to maximize her expected portfolio value at maturity from

trading in the �nancial market so the agent's optimization problem is given by:

supXj∈X

E[uj(−∫ T

0Xjt Ptdt+Hj(PT ) + valuej(Xj

T ))]

. (2.3)

The portfolio value consists of the trading costs −∫ T

0 Xjt Ptdt, the option payo� Hj(PT ) and

the liquidation value valuej(XjT ) of Xj

T stock shares at maturity. Rigorously de�ning a form

of liquidation value in a multi-player framework is a question of its own mathematical interest,

and is not the focus of this paper. The problem of de�ning a liquidation value in a single-player

framework has been solved in a recent paper by Schoeneborn & Schied (2008); in a game-theoretic

setting the problem is much more involved. The agents optimize against their beliefs about the

other players' individual assessments of their respective portfolio values. In order to simplify the

analysis we work under Assumption 2.5 below.

Assumption 2.5. All agents optimize their utility assuming that

valuej(XjT ) =

∫ T

0Xjt Ptdt. (2.4)

The preceding assumption is motivated by the single player, risk-neutral framework where (2.4)

holds in expected values. In such a setting it states that the expected costs (utility) of building

up a portfolio over the time span [0, T ] under market impact equals the expected liquidation

costs (utility) under in�nitely slow liquidation. In a multi-player model Assumption 2.5 is only

a �rst benchmark that simpli�es the subsequent analysis even if all player are risk-neutral. The

assumption is nonetheless useful as it allows us to carry out our analysis of equilibrium and to

derive some insight into the structure of market dynamics under strategic interactions.

Under Assumption 2.5 the individual optimization problem (2.3) reduces to

supXj∈X

E

[uj

(−∫ T

0Xjt g

(N∑i=1

Xit

)dt+Hj(PT )

)]. (2.5)

We say that a vector of strategies(X1, ..., XN

)is a Nash equilibrium if for each agent j ∈ J

her trading strategy Xj is a best response against the behavior of all the other players, i.e., if

6

Xj solves (2.5), given the vector X−j , (Xi)i 6=j . In the following section we derive a-priori

estimates for equilibrium trading strategies and use standard results from the theory of stochastic

optimization to show that Nash equilibria can be characterized in terms of a coupled system of

partial di�erential equations (PDEs). For the special case of risk neutral and CARA investors

we show that the system of PDEs has a solution so that a unique (in a certain class) equilibrium

exists.

3 Equilibrium Dynamics and A-Priori Estimates

In this section we formulate the optimization problem (2.5) as a stochastic control problem, derive

the associated Hamilton-Jacobi-Bellman-equations, HJB for short, and transform it into a system

of coupled PDEs. To this end, we choose the stock price P and the trading costs Rj of the agent

j ∈ J as state variables. They evolve according to:

dPt = σdBt + λN∑i=1

Xitdt, P0 = p0 (3.1)

dRjt = Xjt g

(N∑i=1

Xit

)dt, Rj0 = 0. (3.2)

For a given time t < T , spot price p and trading costs r the value function of the player j, de�ned

by

V j(t, p, r) , supXj∈X

Et

[uj

(−rj −

∫ T

tXjsg

(N∑i=1

Xis

)ds+Hj(PT )

)| Pt = p

], (3.3)

denotes the maximal expected portfolio value at maturity that the player can achieve by trading

the underlying. The associated HJB-equation is0 = vjt +

12σ2vjpp + sup

cj

[λ(cj + X−j

)vjp + cjg

(cj + X−j

)vjrj

]vj(T, p, r) = uj

(−rj +Hj(p)

) (3.4)

cf. Fleming & Soner (1993). Here we have used vjri≡ 0 for i 6= j. Given the trading strategies

X−j of all the other agents, a candidate for the maximizer cj = Xj should satisfy

0 = λvjp

vjrj

+ g(cj + X−j

)+ cjg′

(cj + X−j

)(3.5)

provided vjrj6= 0. We sum up these equations over the set of players in order to get the following

characterization for the cumulated equilibrium trading speed∑N

i=1 Xit :

0 = λN∑i=1

vipviri

+Ng

(N∑i=1

Xit

)+

(N∑i=1

Xit

)g′

(N∑i=1

Xit

). (3.6)

7

Due to Assumption 2.2, z 7→ Ng(z) + zg′(z) is strictly increasing. Hence, equation (3.6) admits a

unique solution X∗ = X∗(∑N

i=1vipviri

). Plugging this solution back into (3.5) allows to compute

the optimal strategy cj = Xj in terms of vip and virias

cj = Xj = − 1

g′(X∗) [λ vjp

vjrj

+ g(X∗)

]. (3.7)

This expression is well de�ned if vjrj6= 0 because g′ > 0. To conclude, we have turned the family

of individual HJB-equations (3.4) into the following system of coupled PDEs for j = 1, ..., N : 0 = vjt +12σ2vjpp + λX∗vjp + Xjg

(X∗)vjrj

vj(T, p, r) = uj(−rj +Hj(p)

) (3.8)

where the coupling stems from the aggregate trading speed X∗ = X∗(∑N

i=1vipviri

).

Solving the system (3.8) is delicate, to say the least. The problem is the non-linearity coming

from the expression Xjg(X∗)along with the implicit dependence of Xj on the derivatives vip

and viriand the fact that vi

riappears in the denominator. The latter problem can be coped with

by assuming that (after a possible monotone transformation) the agents' risk preferences satisfy

a translation property so that ur = 1. A large class of such utility functions can be linked to

backward stochastic di�erential equations but we choose not to embed our work into that line of

research. We consider instead the case of risk neutral and CARA investors where the existence

of a unique classical solution to the system (3.8) is guaranteed without any reference to backward

equations. The proof uses the following a-priori estimates for the optimal trading strategies. It

states that in equilibrium the trading speed is bounded. As a result, the agents' utilities from

trading and the value function associated with their respective HJB equations along with (as we

shall see) their derivatives are bounded.

Lemma 3.1. Let(X1, ..., XN

)be a Nash equilibrium for problem (2.5). Then each strategy Xj

satis�es ∣∣∣Xj∣∣∣ ≤ N λ

εmaxi

∥∥H ip

∥∥∞ .

Proof. Let j ∈ J , h , maxi∥∥H i

p

∥∥∞ and A ,

{∑Ni=1 X

i ≥ 0}

be the set where the aggregate

trading speed is nonnegative. Let us �x the sum of the competitors' strategies X−j . On the

set A the best response Xj is bounded above by K , λεh. Otherwise the truncated strategy

Y j , Xj ∧ λεh would outperform Xj . To see this, let us compare the payo�s associated with Xj

and Y j . The payo� associated with X minus the payo� associated with Y can estimated from

8

below as

−∫ T

0Y jg

(Y j + X−j

)dt+Hj(PT (Y j))

+∫ T

0Xjg

(Xj + X−j

)dt−Hj(PT (Xj))

≥∫ T

0Y j(g(Xj + X−j

)− g

(Y j + X−j

))+(Xj − Y j

)g(Xj + X−j

)dt− λ(Xj

T − YjT ) ‖Hp‖∞ .

Note that Xj + X−j ≥ 0 on A and thus g(Xj + X−j

)≥ 0 due to Assumption 2.2. Furthermore,

g(Xj + X−j

)− g

(Y j + X−j

)≥ ε

(Xj − Y j

), again by Assumption 2.2. The di�erence in the

payo�s is therefore strictly bigger than∫ T

0Y jε

(Xj − Y j

)dt− λh

∫ T

0Xj − Y jdt

≥∫Xj>Y j

(εY j − λh

)(Xj − Y j

)dt

= 0.

This shows that Xj is bounded above on the set A. A symmetric argument shows that on the

complement Ac the optimal response is bounded below by −K. Furthermore, this implies that on

Ac the optimal response is bounded above by

Xj =N∑i=1

Xi +∑i 6=j−Xi ≤ 0 + (N − 1)K

A similar argument yields that Xj is bounded below on A. This completes the proof.

Remark 3.2. These a priori estimates, together with the boundedness assumption on the payo�s

Hj, imply that the optimization problem (2.5) is bounded. Moreover, if a smooth solution to the

PDE system (3.8) exists, it is bounded. �

In the one player framework we can use a standard result from the theory of stochastic control

to show that (3.8) admits a unique solution.

Proposition 3.3. Let N = 1 and for the terminal condition ψ(p, r) , u(−r + H(p)) let ψ ∈ C 3

and let ψ,ψp, ψr satisfy a polynomial growth condition. Then the Cauchy problem (3.8) admits a

unique classical solution in C 1,2, which coincides with the value function V .

Proof. Due to Lemma (3.1), the optimal control X can be chosen from a compact set. Thus, we

can apply Theorem IV.4.3 in Fleming & Soner (1993), which yields that (3.8) admits a unique

solution in C 1,2, which is of polynomial growth. It remains to apply the Veri�cation Theorem

IV.3.1 from Fleming & Soner (1993) to see that this solution coincides with the agent's value

function V .

9

4 Examples

In this section we establish existence and uniqueness of equilibrium results for risk neutral and

CARA investors. For risk neutral investors and linear cost functions the equilibrium strategies can

be given in closed form; if spread crossing costs are involved a closed form solution is not available

and we report numerical results instead.

4.1 Risk Neutral Agents

Let us assume that all players are risk neutral, i.e. uj(z) = z. In a �rst step we prove existence of

a unique solution to the system (3.8) for general cost functions g. Subsequently we construct an

explicit solution to (3.8) for the linear cost structure used in Algmen & Chriss (2001).

4.1.1 General Cost Structure

Let g be a general cost function which satis�es Assumption 2.2. In the risk neutral case the value

function of player j turns into

V j(t, p, r) = −rj + supXj∈X

Et

[−∫ T

tXjsg

(N∑i=1

Xis

)ds+Hj(PT )|Pt = p

].

In particular, V jrj≡ −1 so the optimal strategies do not depend on the trading costs. In other

words, the state variable r is redundant and we omit it in this section. We write

V j(t, p) , V j(t, p, 0).

The HJB-equation (3.4) turns into

0 = vjt +12σ2vjpp + sup

cj

[λ(cj + X−j

)vjp − cjg

(cj + X−j

)](4.1)

where we have used vjrj

= −1. The optimal trading speed form (3.7) is given by

cj = Xj = − 1

g′(X∗) [−λvjp + g(X∗)

](4.2)

where the aggregate trading speed X∗ =∑N

i=1 Xj is the unique solution to

0 = λN∑i=1

vip −Ng

(N∑i=1

Xit

)−

(N∑i=1

Xit

)g′

(N∑i=1

Xit

). (4.3)

The system of PDEs (3.8) therefore takes the form

0 = vjt +12σ2vjpp + λX∗vjp − Xjg

(X∗)

(4.4)

10

with terminal condition vj(T, p) = Hj(p). The following proposition shows that a unique solution

exists if H ∈ C 2b , i.e. H and its derivatives up to order 2 are bounded. The proof follows from a

general existence result stated in Appendix A.

Proposition 4.1. Let H ∈ C 2b . Then the Cauchy problem (4.4) admits a unique classical solution

in C 1,2, which coincides with the vector of value functions.

An alternative way of solving the system (4.4) is the following: If we sum up the N equations,

we get a Cauchy problem for the aggregate value function v ,∑N

i=1 vi, namely

0 = vt +12σ2vpp + X∗

[λvp − g

(X∗)]

with terminal condition v(T, p) =∑N

i=1Hi(p). Existence and uniqueness of a solution to this

one-dimensional problem can be shown using Theorem IV.8.1 in Ladyzenskaja (1968). Once the

solution is known, we can plug it back into (4.4) and get N decoupled equations. This technique

is applied in the following section where we construct an explicit solution for linear cost functions.

4.1.2 Linear Cost Structure

For the particular choice g(z) = κz (κ > 0) used in Algmren & Chriss (2001) and Schoeneborn &

Schied (2007), the solution to (4.4) can be given explicitly.

Corollary 4.2. Let g(z) = κz. Then the solution of (4.4) can be given in closed form as the

solution to a nonhomogeneous heat equation.

Proof. The optimal trading speed from (4.2) and the aggregate trading speed from (4.3) are

Xj =λ

κ

(vjp −

1N + 1

N∑i=1

vip

)(4.5)

X∗ =N∑i=1

Xi =λ

κ(N + 1)

N∑i=1

vip =λ

κ(N + 1)vp. (4.6)

Let us sum up the N equations from (4.4). This yields the following PDE for the aggregate value

function v =∑N

i=1 vi:

0 = vt +12σ2vpp +

λ2N

κ(N + 1)2v2p (4.7)

with terminal condition v(T, p) =∑N

i=1Hi(p). This PDE is a variant of Burgers' equation, cf.

Rosencrans (1972). It allows for an explicit solution, which we cite in Lemma 4.3. With this

solution at hand, we can solve for each single investor's value function. We plug the solution v

back into the equations (4.5) and (4.6) for the trading speeds, and those into the PDE (4.4). This

yields

0 = vjt +12σ2vjpp +

λ2

κ(N + 1)2v2p

11

with terminal condition vj(T, p) = Hj(p). This nonhomogeneous heat equation is solved by

vj(T − t, p) =∫

RHjdN (p, σt) +

λ2

κ(N + 1)2

∫ t

0

∫Rv2p(s, ·)dN (p, σ(t− s))

where v is given in Lemma 4.3 and N denotes the heat kernel.

In the preceding proof and in Corollary 4.5 we need the solution to a variant of Burgers'

equation. We cite it in the following Lemma.

Lemma 4.3. Let A > 0, B 6= 0. The PDE

0 = 2vt +Avpp +Bv2p

with terminal value

v(T, p) = G(p)

is solved by

v(t, p) =A

Blog[∫

Rexp

(B

AG(√

Az))

dN

(p√A, T − t

)].

Proof. By means of a linear transformation we can reduce the problem to A = B = 1. This

particular case is solved in Rosencrans (1972).

4.1.3 Numerical Illustrations

In the risk neutral setting, we were able to reduce the system of PDEs from the multi-player setting

to the one-dimensional PDE (4.7) for the aggregate value function. This can be interpreted as the

value function of the representative agent. Such reduction to a representative agent is not always

possible for more general utility functions. In the sequel we illustrate the optimal trading speed

X(t, p) and surplus3 of a representative agent as functions of time and spot prices for a European

call options H(PT ) = (PT −K)+ and digital option H(PT ) = 1{PT≥K}, respectively. We choose

a linear cost function, strike K = 100, maturity T = 1, volatility σ = 1 and liquidity parameters

λ = κ = 0.01. We see from Figure 1 that for the case of a call option both the optimal trading

speed and the surplus increases with the spot; the latter also increases with the time to maturity.

Furthermore, the increase in the trading speed if maximal, when the option is at the money. For

digital options the trading speed is highest for at the money options close to maturity as the trader

tries to push the sport above the strike. If the spot is far away from the strike, the trading speed

is very small as it is unlikely that the trader can push the sport above the strike before expiry.

3By surplus we di�erence between the representative agent's optimal expected utility v(t, p) and the conditional

expected payo� Et[H(PT )|Pt = p] in the absence of any market impact.

12

Trading Speed

0.0

0.5

1.0

TIME .

90

95

100

105

110

SPOT0.0

0.2

0.4

Surplus

0.0

0.5

1.0

TIME .

90

95

100

105

110

SPOT0.000

0.001

0.002

Figure 1: Trading speed and surplus for a risk neutral investor holding a European Call option.

Trading Speed

0.0

0.5

1.0

TIME .

90

95

100

105

110

SPOT0.0

0.2

0.4

Surplus

0.0

0.5

1.0

TIME .

90

95

100

105

110

SPOT0.0000

0.0005

0.0010

Figure 2: Trading speed and surplus for a risk neutral investor holding a Digital option.

13

98 100 102 104SPOT

0.1

0.2

0.3

0.4

0.5

Trading Speed at t=0

98 100 102 104SPOT

0.0005

0.0010

0.0015

0.0020

0.0025

Surplus at t=0

Figure 3: Trading speed and surplus for a risk neutral investor holding a European Call option for di�erent

spread sizes s = 0 (black), 0.001 (blue), 0.002 (red), 0.003 (green), 0.004 (brown). The higher the spread,

the smaller the trading speed and the surplus.

Figures 3 and 4 illustrate that a high spread makes manipulation unattractive. It shows the

optimal trading speed and the surplus at time t = 0 for the Call and Digital option in the one

player framework. We used the cost function

g(z) = κz + s · sign(z) for di�erent spreads s ∈ {0, 0.001, 0.002, 0.003, 0.004}

with the remaining parameters as above. We see that the higher the spread, the smaller the

trading speed and the surplus. This is intuitive as frequent trading, in particular, when the option

is at the money, incurs high spread crossing costs. The same is true for �xed transaction costs

which also discourage frequent trading.

4.2 Risk Averse Agents

A second class which yields explicit results is those of exponential utility functions uj(z) =− exp

(−αjz

)for j = 1, ..., N , where αj > 0 is the risk aversion coe�cient. In this case the

value functions satisfy

V j(t, p, r) = exp(αjrj

)· V j(t, p, 0)

and thus V jrj

= αjV j . We suppress the state variable r and write V j(t, p) , V j(t, p, 0). As above,we �rst show existence and uniqueness of a solution for a general cost structure. In a second step,

we derive the closed form solution for the linear cost function in the single player framework.

14

98 100 102 104SPOT

0.02

0.04

0.06

0.08

Trading Speed at t=0

98 100 102 104SPOT

0.0002

0.0004

0.0006

0.0008

0.0010

Surplus at t=0

Figure 4: Trading speed and surplus for a risk neutral investor holding a Digital option for di�erent spread

sizes s = 0 (black), 0.001 (blue), 0.002 (red), 0.003 (green), 0.004 (brown). The higher the spread, the

smaller the trading speed and the surplus.

4.2.1 General Cost Structure

The HJB-equation (3.4) turns into

0 = vjt +12σ2vjpp + sup

cj

[λ(cj + X−j

)vjp + cjg

(cj + X−j

)αjvj

](4.8)

with terminal condition vj(T, p) = − exp(−αjHj(p)

). We apply the linear transformation vj ,

− 1αj

log(−vj) to turn the HJB equation into

0 = vjt +12σ2vjpp −

12σ2αj

(vjp)2 + sup

cj

[λ(cj + X−j

)vjp − cjg

(cj + X−j

)](4.9)

with terminal condition vj(T, p) = Hj(p). Note that this equation equals the HJB-equation (4.1)

in the risk neutral setting, up to the quadratic term −12σ

2αj(vjp)2. As in (4.2), the optimal

trading speeds are

cj = Xj = − 1

g′(X∗) [−λvjp + g(X∗)

](4.10)

where the aggregate trading speed X∗ is the unique solution to

0 = λN∑i=1

vip −Ng

(N∑i=1

Xit

)−

(N∑i=1

Xit

)g′

(N∑i=1

Xit

). (4.11)

If we plug X∗ and Xj back into (4.9), we get

0 = vjt +12σ2vjpp −

12σ2αj

(vjp)2 + λX∗vjp − Xjg

(X∗). (4.12)

We can show existence and uniqueness of a solution.

15

Proposition 4.4. Let Hj ∈ C 2b for each j ∈ J . The Cauchy problem 4.9 admits a unique solution,

which coincides with the vector of value functions (up to an exponential transformation).

Proof. See appendix A.

4.2.2 Linear Cost Structure, Single Player

For the one player case with linear cost structure, we have an explicit solution:

Corollary 4.5. Let N = 1 and g(z) = κz. Then the Cauchy problem 4.9 admits a unique solution,

which can be given in closed form.

Proof. The maximizer in (4.9) is

c = X =λ

2κvp

and the Cauchy problem (4.12) turns into

0 = vt +12σ2vpp +

(λ2

4κ− 1

2σ2α

)v2p

with terminal condition v(T, p) = H(p). This is Burgers' equation. Its explicit solution is given

in Lemma 4.3.

4.2.3 Numerical Illustrations

Let us conclude this section with numerical illustrations. We simulated the system (4.8) for two

players. Figure 5 shows the aggregate optimal trading speed X(0, p) + Y (0, p) and the surpluses

vj(0, p) − E[uj (H(PT )) |P0 = p

]for time t = 0 and di�erent spot prices p ∈ [95, 105] for the

European Call option H(PT ) = (PT −K)+; we assume that Player 1 (blue) is the option writer

and Player 2 (red) the option issuer. We chose the strike K = 100, maturity T = 1, volatilityσ = 2 and liquidity parameters λ = κ = 0.01 and risk aversion parameters α1 = 0.01, α2 = 0.01(plain), respectively, α1 = 0.001, α2 = 0.1 (dashed). Since Player 1 has a long position in the

option, she has an incentive to buy the underlying; for the same reason Players 2 has an incentive

to sell it (Panel (b)). Our simulations suggest that the dependence of the equilibrium trading

speed on the agents' risk aversion is weak (Panels (b) and (c)) and that overall the option issuer

is slightly more active than the option writer. Furthermore, we see from Panel (d) that the issuer

bene�ts more from reducing her loss than the writer bene�ts from increasing her gains. This e�ect

is due to the concavity of the utility function and increases with the risk aversion.

16

96 98 102 104SPOT

-1.10

-1.05

-0.95

Value Function at t=0

(a)

96 98 102 104SPOT

-1.0

-0.5

0.5

1.0

Trading Speed at t=0

(b)

96 98 102 104SPOT

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

Cumulated Trading Speed at t=0

(c)

96 98 102 104SPOT

0.0005

0.0010

Surplus at t=0

(d)

Figure 5: Value function, trading speed, aggregate trading speed and surplus for the writer (blue) and

issuer (red) of a European Call option when both agents are risk averse. The plain (dashed) curves display

the case where issuer is about as (more) risk averse than the option writer.

17

5 How to Avoid Manipulation

In this section, we use the closed form solutions for risk neutral agents derived in subsection 4.1.2

to illustrate how an option issuer may prevent other market participants from trading against her

by using their impact on the dynamics of the underlying. Some of our observations were already

made in Gallmeyer & Seppi (2000) in a three-period binomial model. We start with the simplest

case of a zero-sum game.

Corollary 5.1. Let all players be risk neutral with o�setting payo�s∑N

i=1Hi = 0. Then the

aggregate trading speed is∑N

i=1 Xi ≡ 0.

Proof. If∑N

i=1Hi = 0 then (4.6) and (4.7) imply that the aggregate value function v =

∑Ni=1 v

i

equals zero. It follows from (4.6) that the aggregate trading speed vanishes.

Loosely speaking in a zero-sum game, if all option traders are risk neutral and willing to move

the market in their favor, their combined e�ect cancels. We note that this is no longer true for

general utility functions, as illustrated in �gure 5. Of course, in reality market manipulation is

illegal and many investors are unable to manipulate the underlying in the �rst place. This is why

we now look at the following asymmetric situation: The option issuer, Player 0, does not trade the

underlying; her competitor, Player 1, owns the derivative H1 6= 0 and intends to move the stock

price to her favor. In addition, there are N − 1 informed investors without option endowment

in the market. They are �predators� that may supply liquidity and thus reduce the �rst player's

market impact, cf. Carlin, Lobo & Viswanathan (2007) and Schoeneborn & Schied (2007). We

�nd that the more informed competitors are active, the less aggregate manipulation will occur.

Corollary 5.2. Let H1 6= 0 and H i = 0 for i = 2, ..., N . Then limN→∞∑N

i=1 Xit = 0.

Proof. The solution v to the Cauchy problem (4.7) with terminal condition

v(T, p) =N∑i=1

H i(p) = H1(p)

is bounded uniformly in N . Equation (4.6) yields

N∑i=1

Xi =λ

κ

1N + 1

vpN→∞→ 0.

Let us modify the preceding setting a little. Again, Player 0 issues a product H and does not

intend to manipulate the underlying, while her competitors do. More precisely, assume that player

0 splits the product H into pieces and sells them to N risk neutral competitors, such that each of

18

98 100 102 104SPOT

0.1

0.2

0.3

0.4

0.5

cumulated Trading Speed at t=0

(a) Call

98 100 102 104SPOT

0.05

0.10

0.15

0.20

cumulated Trading Speed at t=0

(b) Digital

Figure 6: Aggregate trading speed X∗ at time t = 0 for N = 1 (black), 10 (blue), 100 (red) players each

holding 1/N shares of a Call (left) and Digital (right) option with strike K = 100. The more agents, the

less aggregate manipulation.

them gets 1NH. We �nd that their aggregate trading speed

∑Ni=1 X

i is decreasing in the number

of competitors N . Consequently, the option issuer should sell her product to as many investors as

possible in order to avoid being outsmarted. We illustrate this result in �gure 6, which shows the

aggregate trading speed at time t = 0 of N players each holding 1/N option shares.

Corollary 5.3. Let H i = 1NH for i = 1, ..., N . Then for each t ∈ (0, T ] the aggregate trading

speed∑N

i=1 Xit is decreasing in N and limN→∞

∑Ni=1 X

it = 0.

Proof. As in the preceding corollary, the assertion follows from equations (4.7) and (4.6).

The preceding results indicate how an option issuer can prevent her competitors from manip-

ulation. One strategy is public announcement of the transaction: the more informed liquidity

suppliers on the market, the smaller the impact on the underlying. A second strategy is split-

ting the product into pieces; the more option writers, the less manipulation. Let us conclude this

section with a surprisingly simple way to avoid manipulation: using options with physical delivery.

Remark 5.4. Calls, Puts and Forwards with physical delivery do not induce stock price manipu-

lation. �

Proof. Consider a risk neutral agent who owns Θ > 0 Call options with physical delivery and

strike K. As above, we denote by X her strategy in the underlying. At maturity, she exercises

19

0 ≤ θ ≤ Θ of her Call options. Problem (2.5) turns into:

supX∈X,θ≤Θ

E[∫ T

0−Xt(Pt + κXt)dt+ (XT + θ)

(PT −

12λ(XT + θ)

)− θK

]= sup

X∈X,θ≤ΘE[∫ T

0−κX2

t dt+ θ

(PT −

12λθ

)− θK

]= sup

X∈XE[∫ T

0−κX2

t dt

]+ supθ≤Θ

[θ

(PT −

12λθ

)− θK

]= sup

θ≤Θ

[θ

(PT −

12λθ

)− θK

]where the �rst term in the �rst line describes the expected trading costs in [0, T ] and the liquidationvalue of θ+XT stock shares at maturity. The optimal trading strategy is X ≡ 0. The same holds

true for Put options and Forward options with physical delivery.

6 Conclusion

We investigated the strategic behavior of option holders in illiquid markets. If trading the under-

lying has a permanent impact on the stock price, the possession of derivatives with cash delivery

may induce market manipulation. We showed the existence and uniqueness of optimal trading

strategies in continuous time and for a general cost function; in the one player framework for

general utility functions, and in the multi-player case for risk neutral as well as CARA investors.

Moreover, we showed how market manipulation can be reduced.

Our work may be extended in several directions. Foremost, we derived our results under

Assumption 2.5. This assumption is only satis�ed in the single-player risk-neutral case where the

expected costs of buying a portfolio over a �nite time interval under market impact equals its

expected liquidation value under in�nitely slow liquidation and does not hold in general. The

problem of de�ning a proper notion of liquidation value under strategic interaction is important

but was not our focus and is left for future research. Furthermore, it would be interesting to

consider American or path-dependent options as well as more sophisticated market impact models

such as Obizhaeva & Wang (2006) that account for resilience e�ects and temporary price impacts.

A An Existence Result

In this section, we prove Propositions 4.1 and 4.4 where the PDE (4.4) in the risk neutral setting

is a special case of the system (4.9) for risk averse agents, with αj = 0 for each j. In order to

establish our existence and uniqueness of equilibrium result, we adopt the proof of Proposition

15.1.1 in Taylor (1997) to our framework. After time inversion from t to T − t both systems of

20

PDEs are of the form

vt = Lv + F (vp) (A.1)

for v ,(v1, ..., vN

), where L is the Laplace-operator

L =12σ2 ∂

2

∂p2

and F =(F 1, ..., FN

)is of the form

F j(vp) = −12σ2αj

(vjp)2 + λX∗vjp − Xjg

(X∗).

Here X∗ and Xj are given implicitly by (4.2) and (4.3). The initial condition is

v(0, p) = H(p) =(H1, ...,HN

). (A.2)

We rewrite (A.1) in terms of an integral equation as

v(t) = etL +∫ t

0e(t−s)LF (vp(s))ds , Ψv(t). (A.3)

and seek a �xed point of the operator Ψ on the following set of functions:

X = C 1b (R,RN ) ,

{v ∈ C 1(R,RN ) | v, vp bounded

}equipped with the norm

‖v‖X , ‖v‖∞ + ‖vp‖∞ .

We set Y , Cb. Note that X and Y are Banach spaces and the semi-group etL associated with the

Laplace operator is strongly continuous on X, sends Y on X and satis�es∥∥etL∥∥L (Y,X)

≤ Ct−γ

for some C > 0, γ < 1 and t ≤ 1. Furthermore, the nonlinearity F is locally Lipschitz and

belongs to C∞. Indeed, the map a 7→ X∗(a) is C∞, due to the implicit function theorem with

�rst derivative∂

∂vpX∗(vp) =

λ

(N + 1)g′(X∗(vp)− X∗(vp)g′′(X∗(vp))where the denominator is positive due to Assumption 2.2. The cost function g is C∞ by assump-

tion. In particular, the assumptions of Proposition 15.1.1 in Taylor (1997) are satis�ed.

The a-priori estimates of Proposition 3.1 yield that, if a solution v to (A.1) exists, it is bounded

in the sense ‖v‖X ≤ K. Therefore, we de�ne

XK , {v ∈ X | ‖v‖X ≤ K}

and choosing K large enough we may assume that the initial condition satis�es H ∈ XK .

We are now ready to prove existence and uniqueness of a solution to (A.3). Proposition 15.1.1

in Taylor (1997) gives a solution for a small time horizon [0, τ ], with τ > 0 speci�ed below. We

apply his argument recursively to extend the solution to [0, T ].

21

Proposition A.1. There is τ > 0 such that for each n ∈ N0, the PDE (A.3) with initial condition

(A.2) admits a unique classical, bounded solution in XK on the time horizon [0, nτ ∧ T ]. This

solution coincides with the value function.

Proof. 1. For n = 0, there is nothing to prove. Pick n ∈ N such that nτ < T . By induction,

we can assume that there is a solution v(n) ∈ XK on the time horizon [0, nτ ]. In particular,

the initial condition for the next recursion step h(n) , v(n)(nτ) is in XK .

2. Fix δ > 0. We construct a short time solution on the following set of functions:

Z(n+1) ,{v ∈ C ([nτ, (n+ 1)τ ],X) | v(nτ) = h(n),

∥∥∥u(t)− h(n)∥∥∥

X≤ δ ∀t

}.

To this end, we �rst show that Ψ : Z(n+1) → Z(n+1) is a contraction, if τ > 0 is chosen small

enough.

For this, let τ1 be small enough such that for t ≤ τ1 and any v ∈ XK we have∥∥etLv − v∥∥X ≤12δ.

Here we used that etL is a continuous semigroup and ‖v‖X ≤ K. In particular, for v = h(n):∥∥∥etLh(n) − h(n)∥∥∥

X≤ 1

2δ.

For v ∈ Z(n+1), the derivative vp is uniformly bounded in the sense ‖vp‖∞ ≤∥∥h(n)

∥∥X + δ ≤

K + δ. Hence, we only evaluate F on compact sets. By assumption, F is locally Lipschitz.

In particular, F is Lipschitz on compact sets. In other words, there is a constant K1 such

that for any v, w ∈ Z(n+1) we have

‖F (vp)− F (wp)‖Y ≤ K1 ‖v − w‖X

This implies, for w = h(n)

‖F (vp)‖Y ≤∥∥∥F (h(n)

p )∥∥∥

Y+K1

∥∥∥v − h(n)∥∥∥

X≤ K +K1δ

, K2.

This, together with the boundedness assumption on etL, yields∥∥∥∥∫ t

nτe(t−y)LF (vp(y))dy

∥∥∥∥X≤ t

∥∥etL∥∥ supnτ≤y≤t

‖F (vp(y))‖Y

≤ t1−γCK2.

22

This quantity is ≤ 12δ if t ≤ τ2 ,

(δ

2CK2

) 11−γ

.

Finally, it follows that for v ∈ Z(n+1) we have∥∥∥Ψv − h(n)∥∥∥

X≤

∥∥∥etLh(n) − h(n)∥∥∥

X+∥∥∥∥∫ t

nτe(t−y)LF (vp(y))dy

∥∥∥∥X

≤ 12δ +

12δ = δ.

This shows that Ψ maps Z(n+1) into itself.

It remains to show that Ψ is a contraction. Let v, w ∈ Z(n+1). Then

‖Ψv(t)−Ψw(t)‖X =∥∥∥∥∫ t

nτe(t−y)L [F (vp(y))− F (wp(y))] dy

∥∥∥∥X

≤ t∥∥etL∥∥ sup

nτ≤y≤t‖F (vp(y))− F (wp(y))‖Y

≤ t1−γCK2 supnτ≤y≤t

‖v(y)− w(y)‖X

The quantity t1−γCK2 is ≤ 12 if t ≤ τ3 ,

(1

2CK2

) 11−γ

. This proofs that Ψ is a contraction

in Z(n+1), if τ is small in the sense

0 < τ , min{τ1, τ2, τ3}.

Note that the time step τ does not depend on n. It is the same in every recursion step.

3. It follows that Ψ has a unique �x point v in Z(n+1). In other words, we constructed a

function v ∈ C ([nτ, (n + 1)τ ],X) = C 0,1[nτ, (n + 1)τ ] which solves the PDE (A.3) with

initial condition v(s) = h(n) = v(n)(nτ) on the time interval [nτ, (n+ 1)τ ].

This solution is actually in C 1,2((nτ, (n+ 1)τ ]× R,RN

), due to Proposition 15.1.2 in Taylor

(1997). Furthermore, v is bounded by construction. Indeed, ‖v‖∞ ≤∥∥h(n)

∥∥X + δ ≤ K + δ.

We de�ne the new solution as

v(n+1) , v(n)1{0≤t≤nτ} + v1{nτ<t≤(n+1)τ}.

By construction, v(n+1) solves (A.3) on the time horizon [0, (n+ 1)τ ] and is bounded and in

C 1,2. Hence, we can apply the Veri�cation Theorem IV.3.1 from Fleming & Soner (1993),

which yields that v(n+1) coincides with the vector of value functions (up to time reversal and

an exponential transformation, if αj > 0). The a priori estimate in Proposition 3.1 yields

that v(n+1) ∈ XK . In particular,∥∥v(n+1)((n+ 1)τ)

∥∥X ≤ K, which is necessary for the next

recursion step.

This completes the proof.

23

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25